TELETRAFFIC ENGINEERING

and

NETWORK PLANNING

Villy B. Iversen

COM Course 34340

http://www.com.dtu.dk/education/34340/

Technical University of Denmark

Building 345v

DK2800 Kgs. Lyngby

vbi@com.dtu.dk

Revised January 26, 2007

ii

c Villy Bk Iversen, 2007

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PREFACE

This book covers the basic theory of teletraffic engineering. The mathematical backgroundrequired is elementary probability theory. The purpose of the book is to enable engineers tounderstand ITUT recommendations on traffic engineering, evaluate tools and methods, andkeep up-to-date with new practices. The book includes the following parts:

Introduction: Chapter 1 2,

Mathematical background: Chapter 3 6,

Telecommunication loss models: Chapter 7 11,

Data communication delay models: Chapter 12 14,

Measurements: Chapter 15.

The purpose of the book is twofold: to serve both as a handbook and as a textbook. Thusthe reader should, for example, be able to study chapters on loss models without studyingthe chapters on the mathematical background first.

The book is based on many years of experience in teaching the subject at the TechnicalUniversity of Denmark and from ITU training courses in developing countries.

Supporting material, such as software, exercises, advanced material, and case studies, isavailable at:

http://www.com.dtu.dk/teletraffic

where comments and ideas will also be appreciated.

Villy Bk Iversen

January 26, 2007

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Notations

a Offered traffic per sourceA Offered traffic = AoAc Carried traffic = YA` Lost trafficB Call congestionB Burstinessc ConstantC Traffic congestion = load congestionCn Catalans numberd Slot size in multi-rate trafficD Probability of delay or

Deterministic arrival or service processE Time congestionE1,n(A) = E1 Erlangs Bformula = Erlangs 1. formulaE2,n(A) = E2 Erlangs Cformula = Erlangs 2. formulaF Improvement functiong Number of groupsh Constant time interval or service timeH(k) PalmJacobus formulaI Inverse time congestion I = 1/EJ(z) Modified Bessel function of order k Accessibility = hunting capacity

Maximum number of customers in a queueing systemK Number of links in a telecommuncation network or

number of nodes in a queueing networkL Mean queue lengthLk Mean queue length when the queue is greater than zeroL Random variable for queue lengthm Mean value (average) = m1mi ith (non-central) momentmi ith centrale momentmr Mean residual life timeM Poisson arrival processn Number of servers (channels)N Number of traffic streams or traffic typesp(i) State probabilities, time averagesp{i, t | j, t0} Probability for state i at time t given state j at time t0

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P (i) Cumulated state probabilities P (i) =i

x= p(x)q(i) Relative (non normalised) state probabilities

Q(i) Cumulated values of q(i): Q(i) =i

x= q(x)Q Normalisation constantr Reservation parameter (trunk reservation)R Mean response times Mean service timeS Number of traffic sourcest Time instantT Random variable for time instantU Load functionv VarianceV Virtual waiting timew Mean waiting time for delayed customersW Mean waiting time for all customersW Random variable for waiting timex VariableX Random variabley Carried traffic per souceY Carried trafficZ Peakedness

Carried traffic per channel Offered traffic per idle source Arrival rate for an idle source Palms form factor Lagrange-multiplicatori ith cumulant Arrival rate of a Poisson process Total arrival rate to a system Service rate, inverse mean service time(i) State probabilities, arriving customer mean values(i) State probabilities, departing customer mean values% Service ratio2 Variance, = standard deviation Time-out constant or constant time-interval

Contents

1 Introduction to Teletraffic Engineering 1

1.1 Modelling of telecommunication systems . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 System structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 The operational strategy . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.3 Statistical properties of traffic . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.4 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Conventional telephone systems . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 System structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.2 User behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.3 Operation strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Communication networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 The telephone network . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.2 Data networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.3 Local Area Networks (LAN) . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Mobile communication systems . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4.1 Cellular systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 ITU recommendations on traffic engineering . . . . . . . . . . . . . . . . . . . 16

1.5.1 Traffic engineering in the ITU . . . . . . . . . . . . . . . . . . . . . . . 16

1.5.2 Traffic demand characterisation . . . . . . . . . . . . . . . . . . . . . . 17

1.5.3 Grade of Service objectives . . . . . . . . . . . . . . . . . . . . . . . . 23

1.5.4 Traffic controls and dimensioning . . . . . . . . . . . . . . . . . . . . . 28

1.5.5 Performance monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.5.6 Other recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.5.7 Work program for the Study Period 20012004 . . . . . . . . . . . . . 37

1.5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2 Traffic concepts and grade of service 39

2.1 Concept of traffic and traffic unit [erlang] . . . . . . . . . . . . . . . . . . . . 39

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2.2 Traffic variations and the concept busy hour . . . . . . . . . . . . . . . . . . . 42

2.3 The blocking concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.4 Traffic generation and subscribers reaction . . . . . . . . . . . . . . . . . . . . 48

2.5 Introduction to Grade-of-Service = GoS . . . . . . . . . . . . . . . . . . . . . 55

2.5.1 Comparison of GoS and QoS . . . . . . . . . . . . . . . . . . . . . . . 56

2.5.2 Special features of QoS . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.5.3 Network performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.5.4 Reference configurations . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3 Probability Theory and Statistics 61

3.1 Distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.1.1 Characterisation of distributions . . . . . . . . . . . . . . . . . . . . . 62

3.1.2 Residual lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.1.3 Load from holding times of duration less than x . . . . . . . . . . . . . 67

3.1.4 Forward recurrence time . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.1.5 Distribution of the jth largest of k random variables . . . . . . . . . . 69

3.2 Combination of random variables . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.2.1 Random variables in series . . . . . . . . . . . . . . . . . . . . . . . . 70

3.2.2 Random variables in parallel . . . . . . . . . . . . . . . . . . . . . . . 71

3.3 Stochastic sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 Time Interval Distributions 77

4.1 Exponential distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1.1 Minimum of k exponentially distributed random variables . . . . . . . 79

4.1.2 Combination of exponential distributions . . . . . . . . . . . . . . . . 80

4.2 Steep distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3 Flat distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3.1 Hyper-exponential distribution . . . . . . . . . . . . . . . . . . . . . . 83

4.4 Cox distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4.1 Polynomial trial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4.2 Decomposition principles . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4.3 Importance of Cox distribution . . . . . . . . . . . . . . . . . . . . . . 90

4.5 Other time distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.6 Observations of life-time distribution . . . . . . . . . . . . . . . . . . . . . . . 92

5 Arrival Processes 95

5.1 Description of point processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

CONTENTS ix

5.1.1 Basic properties of number representation . . . . . . . . . . . . . . . . 97

5.1.2 Basic properties of interval representation . . . . . . . . . . . . . . . . 98

5.2 Characteristics of point process . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2.1 Stationarity (Time homogeneity) . . . . . . . . . . . . . . . . . . . . . 100

5.2.2 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2.3 Simple point process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.3 Littles theorem . . . . . . . . . .